[I really need to return to blogging. My lack of focus on reflection has hampered me this semester, and I need to fix that. To that end, I am making a commitment to blog and to jog. Those are the foci this year of the ellipse that is my world.]

Yea, how often does that happen that a class gets excited about logs? It has not happened to me in several years, but this year I found a way. We started the second semester with graphing again. We have a standard list of things we look for, identify, and document on every single graph. The list is:

Domain:

Range:

Asymptototes (vertical and / or slant):

Minimums:

Maximums:

Vertex:

Y-intercept:

X-intercept:

End state behavior:

Every graph we do, we have to document all of these items. If we graph a line, most of the list is “none” but it creates the connection between all the graphs. Every graph has the same questions, it is just that some of the graphs / functions do not have those features.

So, I am doing this file on Desmos, and we are documenting. They have done all these as homework, so really we are checking answers and ensuring learning. Then weird things happen. They notice the symmetry of the inverses.

Nice.

Then they ask to see the graph of the line of symmetry. Even nicer. THEN! OMG. We put the translation into the h-k form of the line, and we see the translation of the line of symmetry.  [Okay, seriously. If you are not using the h-k forms to make connections, why not. See This post, or This post or any other of the several posts I have on this topic.]

And then I graph the exponential. …. …. They know there must be an inverse, but nothing we have done in class looks like that. …. And then, because I have the list of all the h-k forms on the board, someone asks, “Is that what the log thingy is for?”

And now they have a reason to learn logs. They are intrigued by logs. They are asking questions about logs. Because EVERYTHING in math has a forwards and a backwards, addition has subtraction, squares have square roots, and exponentials have logarithms.

They are interested and inquisitive about a topic that normally is not approached this way. I have done something good I think. Only time will tell if I can continue that on this topic.

Yea, that’s right. I just had 2 classes in a row teach themselves and others how to complete the square with circles and ellipses. How did I accomplish this miracle, because I really do consider it to be a miracle.

In my Advanced Algebra Class we have the following problem called, creatively enough, The Lost Hiker.

Suppose you need to find yet another lost hiker.  Fortunately you have information from 3 different radio transmitters.  From this information you know that he is:

25 miles from Transmitter A

5 miles from Transmitter B and

13 miles from Transmitter C

These radio transmitters are at the following coordinates:

Transmitter A – (25.5, 7.5)

Transmitter B – (-2.5, 3.5)

Transmitter C – (6.5, -11.5)

Use this information to find the coordinates of the lost hiker using the intersections of the theoretical circles.  Show your work below.

There are 4 major steps here; 1. Write the equation of 3 circles, 2. Expand the circle equations, 3. “Collapse” the circles through polynomial subtraction to 2 lines, and 4. Solve the system of lines.

We worked on these for 3 days. These problems are huge and complex, and one single minus sign incorrect means the whole thing works out wrong. But they persevered for three periods in class, until every single learner could do 3 unique problems on their own, and get the correct answer.

I have an excel spreadsheet programmed to calculate an infinite number of nice problems with solutions. The solutions are posted on the board w/ a magnet, and I walk around with a bunch of problems in my hand. Get one right, here is another.

Well, at the end of 3 days, I wrote the following equation on the board and asked them to turn it back into a circle.

x2+ y2 – 6x + 8y = 12

It took them all of 1 minute to have the entire class finished. One learner who failed Alg2 taught the rest how to do it because it was, “Easy, just take the -6, divide by two, because when you square it you have two of them.”

Then I wrote 5 problems on the board.

x2+ y2 – 4x + 12y = 10

x2+ y2 + 2x – 10y = -8

x2+ y2 – 4x + 7y = -20

2x2+ 3y2 + 6x + 15y = 0

4x2+ 5y2 + 16x – 100y = 27

The first two were just some more practice. The question was asked, “Can we have a decimal?” When the answer was yes, they didn’t bat an eye.

The last two did make them think, but they had the factoring done correctly on the left side, they just needed a little hinting for the right side of the equals.

Why did it work so nicely? I think because they really were engaged with the lost hiker problem and honestly worked them. And they worked. They agonized over why they didn’t get the correct answer. They gave me dirty looks, and when we found the minus sign they missed or the extra number they wrote in, they were angry at themselves.

But they persisted! It was a thing of beauty and I loved it.

When they did the completing the square on their own, with only some gentle nudging from me, I told them how proud of them I am. They needed to hear it.

Heck, they EARNED it.

My AP Stat class has been frustrating me the last couple of weeks. They have been very blah. Not interested in being aggressive in their learning, not thinking about the formulas for stats, and in general willing to just sit there and expect the knowledge to jump into their heads with no effort.

It came to a head last week where we did a problem as a warm-up, then did two problems from homework, and when I handed out an AP FRQ, they all looked at me as if it was the first time they had ever seen the problem when in reality we had just did three of the identical problems prior.

Yea, I was frustrated.

But, instead of freaking out, I started asking them questions, mainly, “Why?” It turns out they were feeling overwhelmed and frustrated too. The formula sheet was a complete mystery to them, the concepts where going over their heads, and they wanted to freak out as well.

So I burned a day going over the formula sheet.

Guess what, no one has ever taught them how to understand what is on the formula sheet. That should not be a surprise to me, but it is. So here is a sample of what I did:

7 elements in the definition of mean are; xbar, equals, sigma, x, i, n, and divide by. I made them explain what each was, and the fact that the equals sign can mean different things was the toughest. So was the fact that the ‘1’s’ in standard deviation are different. We really discussed what these things meant.

Each formula / equation / definition has the number of elements indicated, and a “Hand or Calc?” statement next to it. I didn’t want them to understand the formula and then forget they are supposed to do standard deviation in their calc not by hand. But they need to understand the formula too.

No one has ever done this with them in any class before, this was the first time they have had to really understand a formula sheet. They hated it. They struggled, they discussed, they asked questions, they growled and shot me dirty looks, and EVERY SINGLE person said it helped them understand the formulas better. This, in turn, means they understand the class better.

Success is a beautiful thing when it works. This worked.

Here is the docx if you want it. It is formatted for use in Interactive Notebooks.

A little background before I explain what was said to me.

I was part of an IREX grant program that brings teachers from other countries to the US to learn about education, etc. My part of this was to host a teacher from Jordan for 6 weeks in my classroom 2 times per week, and mentor him on mathematics education, technology, etc.

It was a great program, and I learned a lot from interacting with Mr. S. and I think we will built a relationship that will continue in the future for the benefit of us both.

With that said, on his last day in my classroom he told me something that really changed how I viewed myself and my classroom behavior.

He said (and I am paraphrasing here) that in 2008 he wrote an essay for a teaching award in Jordan and in the essay he said he wanted to be the kind of teacher that leads learners to the result, but doesn’t tell them how to get there. He wanted to be the kind of teacher that creates problems for the learners to solve, and lets them work through the problems together, and the learn along the way how to solve other problems because of that. And then he said he wanted to be the kind of teacher that has a mutual respect in the classroom that allows for a good exchange of ideas between learners and teacher.

I said, of course, I think we all want to be that teacher. To which he dropped the largest complement I think I could ever get.

Mr. S. then said, “I wrote that 4 years ago, and I have never seen that kind of classroom until I had the opportunity to sit in yours. You are that teacher I said I wanted to be several years ago.”

You know, it took me several hours of thinking of that to really realize just what he said and what it means to me. I don’t think I have ever had someone say anything so nice, so supportive and so mentally Earth shattering to me.

I have never considered myself to be the teacher he described, but clearly this veteran teacher from Jordan, with many years of experience, and national awards in his own country thinks I am. And he thinks it because he spent 2 days a week, for 6 weeks in my classroom.

How do I live up to that? It still shocks me, and it was 4 days ago he said it. I guess I have never thought of myself that way, but maybe I should start. So I can grow, develop and continue to improve.

Or does that improvement come because I DON’T think it is true? I am not sure, but I am still blown away.

Thank you Mr. S. for making me think of myself in a very new way. I hope I can return the favor someday, because you have impacted me tremendously.

I am rather late to the gate with these thoughts, in large part because I drove (well, rode my motorcycle) to St. Louis from Reno, which took 4 days to get home. More on that in a different post when I do the math from the trip video.

If you ask one of the teachers from my department about me, one thing almost all of them will say or agree with is that I hate paper. I hate paper note, paper suggestions, paper anything. I ask my department to submit everything to me through email, because I lose paper notes, throw away binders, and just downright hate keeping track of paper.

With that said, I admit I took 11 pages of notes, kept track of them through 4 days of a conference, and 2000 miles of motorcycle riding. This event meant that much to me, and that is saying a lot.

I am going to start on day 1 and work through all my notes. I am composing this more for myself than anyone else. Like all of my reflective posts, it really doesn’t matter if anyone else ever reads them. This is the place where I reflect and compose my thoughts for my future self (because if I do it on paper, it gets thrown away!)

Wednesday – A personal exploration:

To say I was nervous on Wednesday would be an understatement. I had just ridden 1900 miles over three days to meet with a group of people I had never met and only talked to through twitter, emails and blogs. Yea, I was nervous. You see, I am an incredible introvert. I attended a NCTM conference my first year as a teacher with 2 other teachers from my school. At the end of the conference, I had spoken with exactly 3 people in a meaningful way, 2 of which came with me. I attended sessions and sat in the back. I wandered the floor and just kind of nodded and said the minimum I had to.

At that point I realized that if I was going to become a half way decent teacher, I had to overcome this fear of putting myself out there. Even riding my motorcycle to the event is a sign of my hesitance to talk to people. I was riding alone, where the only conversation going on was inside my helmet. If I flew I might have to actually talk to someone (although I usually don’t when I fly, either.) This has been my most difficult challenge as a teacher, because it is so incredibly easy to just stay in my room and never reach out. I used to actually have it on my to do list every week, “Make contact with other teachers.” I don’t any more, but the tendencies remain.

So I walked into the hotel dirty and sweaty from riding in the heat and the first thing that happens is Lisa H. tries to give me a hug! Okay, ice broken, I can deal and grow.

After a shower so I was feeling like a human being again, a group of us walked up to the nearby mall and had a nice dinner at a southwestern restaurant and chatted. It was nice, and it gave me a chance to get to know people in a very casual way. It definitely was the right thing to do. I did think about staying in the hotel and sleeping, but decided otherwise. I am glad I did.

The actual events of the week after the break (warning it is long & detailed):

First off, I am a math teacher, so I think it goes without saying I am incredibly … detail oriented … anal … and, possibly, a pain in the butt (depending on when you ask my wife.)

I absolutely have to save every single receipt from vacation, reconcile with Quicken within 10 hours of arriving home on vacation, and that reconciliation must be to the penny. Driving to Montana? Great, that means 7 gas receipts, organized by location, along with the mileage at every gas stop.

Hotels? They give a full page receipt, not a little strip like gas stations. Dining out? Who was there? Where was the restaurant? What city?

Okay, you get my drift. I like to keep good records of our vacations. This used to be a major chore. My wife said this was not necessary, but to me, those little details hid some nice surprises. For instance, in our 11 year old car, that now has over 100000 miles we are still getting over 33 miles per gallon on the freeway. If I didn’t take good records, how would I know that! See, it is important!

Enter Evernote. When I first heard of Evernote, I was not that interested. A note taking software for my phone. Whatever. On the computer too! Yawn. I use OneNote.

But, I had a recently thought to myself, “What if I took a picture of the receipts as I traveled?” That would mean I don’t have to keep them. [After all, these are vacation receipts, not store receipts. I won’t need them to return merchandise, so this won’t apply.]

So I did that on our most recent trip to MT. Here is what it looks like.

[Click the pic to embiggen]

Notice that if I don’t put in a title when I click “New Note” it GeoTags my picture! Now I have a running record of my location by date! Love it. At each stop, I also put in the milage in the car. Now I can easily calculate the mpg.

This also solves some additional problems. In Missoula the machine was out of paper. I did not need to go in and ask for a receipt. All I needed to do is take a pic of the gas pump. It automatically captured the location, and all I really needed was purchase price and gallons anyway.

[Click the pic to embiggen]

By clicking the “Show Details” or “Hide Details” in the upper right you can see additional info, and by clicking the Map Icon or city name above the pic, you can view a Google map of exactly where you were.

[Click the pic to embiggen]

Now I am thinking of much better uses for this information. I used it only for receipts this time, but what if I also included neat locations along the way? Now I am kicking myself for not having my phone out and Evernote open when we had the herd of 6 bighorn sheep in the road with us. (And no, I don’t have ANY photos of that. Dang.)

The only drawback I saw was it was slow. Evernote was communicating with the camera and the GPS all at once. I don’t think I would record a whole vacation in there yet. My phone needs to be faster, and the cell connection needs to be much faster (still only on 3G) before it would get fast enough and error proof enough to make me trust it completely.

But, I came home from vacation with no receipts bulging in my wallet and stuffed in my suitcase. I didn’t need to spread them all out on the desk and examine them for dates to put them in order. All I did was fire up the computer, log into Evernote, and go down the list.

It saved me about an hour in my data entry, and actually made my life much easier. Now, how can I leverage this in my classroom? Hmmmm.

Something to think about on my next trip. Which is tomorrow. Must go pack.

My AP Stat learners had a project for their final exam, it was very straightforward. In our area, we have 4 grocery stores.

If you click the image, you will see the larger image that shows on 2 exits of the freeway the four stores (Scolari’s is a local chain) and on the far right of the map, our school. Walmart opened up only 3 months ago, so it is the new kid on the block, and of course, their marketing screams at us every day that they have the cheapest prices on groceries.

Our question was, do they really have the cheapest prices? The rest after the break.

Today I tried something I have never done before. I used a “chain note” activity as a pre-learning exercise. In case you don’t know what a chain note is, I will explain.

A chain note is a note started by one person, and continued by another. In this case, I had an envelope with an essential learning on the outside. I numbered off the class into groups, in this case 1 through 6, and each group received an empty envelope to begin. The group answered the question, and after 4 minutes (that is all I gave them to start) they rotated envelopes. The new group had to answer the question, and then revise/ comment on/ or correct the answer in the envelope.

By the last rotation, they were answering, then examining 5 other answers and seeing what they could contribute to each of the other answers.

The best part is that I get to read their answers, so now I know what they really understand and what they are still struggling with. I like that. When I now formally introduce the chapter on Z-Tests, I know what I can skim over, and where I really need to slow down.

Very useful stuff!I know that I better review the difference between a standard deviation and standard error (from last chapter), and I know that the fact that null hypothesis are what we are testing is pretty well understood.

I recommend this strongly, and I will be doing it again. In case anyone is interested, here is version 1 of the prompts I used. I know these will change greatly for version 2 next year.

Define “Null Hypothesis, H0” in your own words, and give an example.

Define “Alternative Hypothesis, HA” in your own words, and give an example.

What is the difference between Standard Deviation and Standard Error?

What does “p-value” mean? Are you looking for large or small values?

What does the “p-value” represent?

What conditions do you need to check EVERY time?